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14-September-2008 18:02:38 - P-value In statistical hypothesis testing, the p-value is the probability of obtaining a result at least as extreme as the one that was actually observed, given that the null hypothesis is true. The fact that p-values are based on this assumption is crucial to their correct interpretation. More technically, a p-value of an experiment is a random variable defined over the sample space of the experiment such that its distribution under the null hypothesis is uniform on the interval 0,1. Many p-values can be defined for the same experiment. Contents 1 Coin flipping example 2 Interpretation 3 Frequent misunderstandings 4 See also 5 Additional reading 6 References 7 External links Coin flipping example For example, say an experiment is performed to determine if a coin flip is fair 50% chance of landing heads or tails, or unfairly biased, either toward heads 50% chance of landing heads or toward tails 50% chance of landing heads. Since we consider both biased alternatives, a two-tailed test is performed. The null hypothesis is that the coin is fair, and that any deviations from the 50% rate can be ascribed to chance alone. Suppose that the experimental results show the coin turning up heads 14 times out of 20 total flips. The p-value of this result would be the chance of a fair coin landing on heads at least 14 times out of 20 flips plus the chance of a fair coin landing on heads 6 or fewer times out of 20 flips. In this case the random variable T has a binomial distribution. The probability that 20 flips of a fair coin would result in 14 or more heads is 0.0577. By symmetry, the probability that 20 flips of the coin would result in 14 or more tails alternatively, 6 or fewer heads is the same, 0.0577. Thus, the p-value for the coin turning up heads 14 times out of 20 total flips is 0.0577 + 0.0577 = 0.1154 . Interpretation Generally, one rejects the null hypothesis if the p-value is smaller than or equal to the significance level, often represented by the Greek letter α alpha. If the level is 0.05, then the results are only 5% likely to be as extraordinary as just seen, given that the null hypothesis is true. In the above example we have: null hypothesis H0 - fair coin; observation O - 14 heads out of 20 flips; and probability p-value of observation O given H0 - pO|H0 = 0.0577x2 two-tailed = 0.1154 percentage expressed as 11.54%. The calculated p-value exceeds 0.05, so the observation is consistent with the null hypothesis - that the observed result of 14 heads out of 20 flips can be ascribed to chance alone - as it falls within the range of what would happen 95% of the time were this in fact the case. In our example, we fail to reject the null hypothesis at the 5% level. Although the coin did not fall evenly, the deviation from expected outcome is just small enough to be reported as being not statistically significant at the 5% level. However, had a single extra head been obtained, the resulting p-value two-tailed would be 0.0414 4.14%. This time the null hypothesis - that the observed result of 15 heads out of 20 flips can be ascribed to chance alone - is rejected. Such a finding would be described as being statistically significant at the 5% level. Critics of p-values point out that the criterion used to decide statistical significance is based on the somewhat arbitrary choice of level often set at 0.05. A proposed replacement for the p-value is p-rep. It is necessary to use a reasonable null hypothesis to assess the result fairly. The choice of null hypothesis entails assumptions. Frequent misunderstandings The conclusion obtained from comparing the p-value to a significance level yields two and three results: either the null hypothesis is rejected, or the null hypothesis cannot be rejected at that significance level. You cannot accept the null hypothesis simply by the comparison just made 11% 5%; there are alternative tests that have to be performed, such as some goodness of fit tests. It would be very irresponsible to conclude that the null hypothesis needs to be accepted based on the simple fact that the p-value is larger than the significance level chosen. The use of p-values is widespread; however, such use has come under heavy criticism due both to its inherent shortcomings and the potential for misinterpretation. There are several common misunderstandings about p-values.1 2 The p-value is not the probability that the null hypothesis is true claimed to justify the rule of considering as significant p-values closer to 0 zero. In fact, frequentist statistics does not, and cannot, attach probabilities to hypotheses. Comparison of Bayesian and classical approaches shows that a p-value can be very close to zero while the posterior probability of the null is very close to unity. This is the Jeffreys-Lindley paradox. The p-value is not the probability that a finding is merely a fluke again, justifying the rule of considering small p-values as significant. As the calculation of a p-value is based on the assumption that a finding is the product of chance alone, it patently cannot simultaneously be used to gauge the probability of that assumption being true. This is subtly different from the real meaning which is that the p-value is the chance that null hypothesis explains the result: the result might not be merely a fluke, and be explicable by the null hypothesis with confidence equal to the p-value. The p-value is not the probability of falsely rejecting the null hypothesis. This error is a version of the so-called prosecutor's fallacy. The p-value is not the probability that a replicating experiment would not yield the same conclusion. 1 - p-value is not the probability of the alternative hypothesis being true see 1. The significance level of the test is not determined by the p-value. The significance level of a test is a value that should be decided upon by the agent interpreting the data before the data are viewed, and is compared against the p-value or any other statistic calculated after the test has been performed. The p-value does not indicate the size or importance of the observed effect compare with effect size. See also Counternull Statistical hypothesis testing Additional reading Dallal GE 2007 Historical background to the origins of p-values and the choice of 0.05 as the cut-off for significance Hubbard R, Armstrong JS 2005 Historical background on the widespread confusion of the p-value PDF Fisher's method for combining independent tests of significance using their p-values Dallal GE 2007 The Little Handbook of Statistical Practice A good tutorial References ^ Sterne JAC, Smith GD 2001. Sifting the evidence - what's wrong with significance tests?. BMJ 322 7280: 226-231. doi:10.1136/bmj.322.7280.226. PMID 11159626. ^ Schervish MJ 1996. P Values: What They Are and What They Are Not. The American Statistician 50 3: 203-206. External links Free online p-values calculators for various specific tests chi-square, fisher's F-test, etc. Understanding P-values, including a Java applet that illustrates how the numerical values of p-values can give quite misleading impressions about the truth or falsity of the hypothesis under test. v d e Statistics Experimental design Design of experiments - Sampling - Stratified sampling - Replication - Blocking - Statistical power - Sample size Descriptive statistics Continuous data Mean Arithmetic, Geometric, Harmonic - Median - Mode - Range - Variance - Standard deviation - Skewness - Kurtosis - Effect size Categorical data Frequency - Contingency table Inferential statistics Bayesian inference - Hypothesis testing - Significance - Power - Null hypothesis/Alternative hypothesis/Error - Z-test - Student's t-test - Chi-square test - F-test - P-value - Interval estimation - Maximum likelihood - Analysis of variance - Meta-analysis - Confidence interval Survival analysis Survival function - Kaplan-Meier - Logrank test - Failure rate - Proportional hazards models Correlation Confounding variable - Pearson product-moment correlation coefficient - Rank correlation Spearman's rank correlation coefficient, Kendall tau rank correlation coefficient Regression analysis Linear regression - Nonlinear regression - Logistic regression - Maximum Likelihood Estimation Statistical graphics Bar chart - Biplot - Box plot - Control chart - Histogram - Q-Q plot - Run chart - Scatterplot - Stemplot Category · Project · Portal · List of topics Retrieved from http://en..org/wiki/P-value Categories: Hypothesis testing Views Article Discussion this page History Personal tools Log in / create account Navigation Main page Contents Featured content Current events Random article Search Go Search Interaction Community portal Recent changes Contact Donate to Help Toolbox What links here Related changes Upload file Special pages Printable version Permanent link Cite this page Languages العربية Deutsch Español Ù?ارسی Français Italiano Nederlands Polski Português Basa Sunda اردو This page was last modified on 11 September 2008, at 21:0
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